Given a social network with diffusion probabilities as edge weights and an integer k, which k nodes should be chosen for initial injection of information to maximize influence in the network? This problem is known as Target Set Selection in a social network (TSS Problem) and more popularly, Social Influence Maximization Problem (SIM Problem). This is an active area of research in computational social network analysis domain since one and half decades or so. Due to its practical importance in various domains, such as viral marketing, target advertisement, personalized recommendation, the problem has been studied in different variants, and different solution methodologies have been proposed over the years. Hence, there is a need for an organized and comprehensive review on this topic. This paper presents a survey on the progress in and around TSS Problem. At last, it discusses current research trends and future research directions as well.